# Library for sparse tensorized polynomials

**URL:** <https://discourse.julialang.org/t/library-for-sparse-tensorized-polynomials/101549>\
**Category:** General Usage\
**Tags:** question\
**Created:** [July 12, 2023, 7:35pm UTC](https://discourse.julialang.org/t/library-for-sparse-tensorized-polynomials/101549 "2023-07-12T19:35:41Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![mleprovost](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mleprovost/32/7166_2.png) [@mleprovost](https://discourse.julialang.org/u/mleprovost)\
**Post date:** [July 12, 2023, 7:35pm UTC](https://discourse.julialang.org/t/library-for-sparse-tensorized-polynomials/101549/1 "2023-07-12T19:35:42Z")

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Hello,

I am wondering if there is a library for sparse tensorized poynomials?

For x \in \mathbb{R}^n, I am interested in functions f:\mathbb{R}^n \to \mathbb{R} that can be decomposed as a sum of tensorized polynomials:  
f(x) = \sum\_{\boldsymbol{\alpha}} c\_{\boldsymbol{\alpha}} \psi\_{\boldsymbol{\alpha}}(\mathbf{x}), where \boldsymbol{\alpha} = (\alpha\_1, \ldots, \alpha\_n) is a multi-index of non-negative integers and \psi\_{\boldsymbol{\alpha}}(\mathbf{x}) = \prod\_{j=1}^n \phi\_{\alpha\_j}(x\_j), where \{ \phi\_i \}\_{i \in \mathbb{N} } is a family of univariate polynomials indexed over \mathbb{N}, e.g. univariate Hermite polynomials. Typically the multi-index \boldsymbol{\alpha} will be sparse.

I am particularly interested in fast computations of quantities like \nabla\_x f(x) or \nabla\_{c\_{\boldsymbol{\beta}}} f by leveraging sparsity of the multi-indices \boldsymbol{\alpha}'s.

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**Author:** ![cmarcotte](https://avatars.discourse-cdn.com/v4/letter/c/a3d4f5/32.png) [@cmarcotte](https://discourse.julialang.org/u/cmarcotte)\
**Post date:** [July 13, 2023, 8:18am UTC](https://discourse.julialang.org/t/library-for-sparse-tensorized-polynomials/101549/2 "2023-07-13T08:18:13Z")

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[AdaptiveSparseGrids.jl](https://github.com/jacobadenbaum/AdaptiveSparseGrids.jl) and [DistributedSparseGrids.jl](https://github.com/baxmittens/DistributedSparseGrids.jl) both implement Smolyak grids for sparse approximation of functions over high-dimensional domains. These might be useful for you.